Abstract
In this paper we study some problems of the canonical harmonic analysis on the field \({\mathbb Q}_p\) of \(p\) -adic numbers. The main elements of the canonical harmonic analysis on \({\mathbb Q}_p\) are canonical Fourier integral transforms, canonical generalized translation operators and canonical convolution products for functions on \({\mathbb Q}_p\) . We consider various results of the canonical harmonic analysis for functions from Lebesgue spaces \(L^\rho({\mathbb Q}_p)\) , \(1\le\rho\le\infty\) . Basic concepts of the canonical harmonic analysis on \({\mathbb Q}_p\) are expand to generalized functions (or distributions), among them the canonical Fourier transforms on \({\mathbb Q}_p\) , the generalized translation operators on \({\mathbb Q}_p\) and others. The analogues of various results of classical harmonic analysis, including analogues of the Paley-Wiener-Schwartz theorems, are proved. We introduce a canonical convolution product on \({\mathbb Q}_p\) for usual and generalized functions and establish some of its properties.
